A337434 Essentially different n X n matrices with maximal sum of entries satisfying the conditions of A228882.
List of solutions for n=5 and n=6.

A228882(5) = 63; 20 distinct solutions:
 [1,2,1,2,3;4,3,4,5,1;2,1,2,3,4;3,4,5,1,2;1,2,3,1,3]
 [1,2,1,2,3;4,3,4,5,1;2,1,2,3,4;3,5,4,1,2;1,2,3,1,3]
 [1,2,1,2,3;4,3,5,4,1;2,1,2,3,4;3,4,5,1,2;1,2,3,1,3]
 [1,2,1,2,3;4,3,5,4,1;2,1,2,3,4;3,5,4,1,2;1,2,3,1,3]
 [1,2,3,4,1;3,4,1,2,3;2,5,3,4,1;1,1,2,5,3;2,3,4,1,2]
 [1,2,3,4,1;3,4,1,2,3;2,5,3,4,1;1,1,2,5,3;2,4,3,1,2]
 [1,2,3,4,1;3,4,1,2,3;2,5,3,5,1;1,1,2,4,3;2,3,4,1,2]
 [1,2,3,4,1;3,4,1,2,3;2,5,3,5,1;1,1,2,4,3;2,4,3,1,2]
 [1,2,3,4,1;3,5,1,2,3;2,4,3,5,1;1,1,2,4,3;2,3,4,1,2]
 [1,2,3,4,1;3,5,1,2,3;2,4,3,5,1;1,1,2,4,3;2,4,3,1,2]
 [1,2,4,3,1;3,4,1,2,3;2,5,3,4,1;1,1,2,5,3;2,3,4,1,2]
 [1,2,4,3,1;3,4,1,2,3;2,5,3,4,1;1,1,2,5,3;2,4,3,1,2]
 [1,2,4,3,1;3,4,1,2,3;2,5,3,5,1;1,1,2,4,3;2,3,4,1,2]
 [1,2,4,3,1;3,4,1,2,3;2,5,3,5,1;1,1,2,4,3;2,4,3,1,2]
 [1,2,4,3,1;3,5,1,2,3;2,4,3,5,1;1,1,2,4,3;2,3,4,1,2]
 [1,2,4,3,1;3,5,1,2,3;2,4,3,5,1;1,1,2,4,3;2,4,3,1,2]
 [1,3,1,3,2;3,2,4,5,1;4,1,3,2,3;2,5,4,1,4;1,3,2,1,2]
 [1,3,1,3,2;3,2,4,5,1;4,1,3,2,4;2,5,4,1,3;1,3,2,1,2]
 [1,3,1,3,2;4,2,4,5,1;3,1,3,2,3;2,5,4,1,4;1,3,2,1,2]
 [1,3,1,3,2;4,2,4,5,1;3,1,3,2,4;2,5,4,1,3;1,3,2,1,2]

A228882(6) = 93; 64 distinct solutions:
 [1,1,2,4,1,3;2,4,5,3,1,2;4,3,1,2,3,4;1,2,3,4,5,1;3,4,5,1,2,3;2,1,2,3,4,1]
 [1,1,2,4,1,3;2,4,5,3,1,2;4,3,1,2,3,4;1,2,3,4,5,1;3,4,5,1,2,3;2,1,2,4,3,1]
 [1,1,2,4,1,3;2,4,5,3,1,2;4,3,1,2,3,4;1,2,3,4,5,1;3,5,4,1,2,3;2,1,2,3,4,1]
 [1,1,2,4,1,3;2,4,5,3,1,2;4,3,1,2,3,4;1,2,3,4,5,1;3,5,4,1,2,3;2,1,2,4,3,1]
 [1,1,2,4,1,3;2,4,5,3,1,2;4,3,1,2,3,4;1,2,3,5,4,1;3,4,5,1,2,3;2,1,2,3,4,1]
 [1,1,2,4,1,3;2,4,5,3,1,2;4,3,1,2,3,4;1,2,3,5,4,1;3,4,5,1,2,3;2,1,2,4,3,1]
 [1,1,2,4,1,3;2,4,5,3,1,2;4,3,1,2,3,4;1,2,3,5,4,1;3,5,4,1,2,3;2,1,2,3,4,1]
 [1,1,2,4,1,3;2,4,5,3,1,2;4,3,1,2,3,4;1,2,3,5,4,1;3,5,4,1,2,3;2,1,2,4,3,1]
 [1,1,2,4,1,3;2,4,5,3,1,2;4,3,1,2,4,3;1,2,4,3,5,1;4,3,5,1,2,3;2,1,2,3,4,1]
 [1,1,2,4,1,3;2,4,5,3,1,2;4,3,1,2,4,3;1,2,4,3,5,1;4,3,5,1,2,3;2,1,2,4,3,1]
 [1,1,2,4,1,3;2,4,5,3,1,2;4,3,1,2,4,3;1,2,5,3,5,1;4,3,4,1,2,3;2,1,2,3,4,1]
 [1,1,2,4,1,3;2,4,5,3,1,2;4,3,1,2,4,3;1,2,5,3,5,1;4,3,4,1,2,3;2,1,2,4,3,1]
 [1,1,2,4,1,3;2,4,5,3,1,2;4,3,1,2,5,3;1,2,4,3,4,1;4,3,5,1,2,3;2,1,2,3,4,1]
 [1,1,2,4,1,3;2,4,5,3,1,2;4,3,1,2,5,3;1,2,4,3,4,1;4,3,5,1,2,3;2,1,2,4,3,1]
 [1,1,2,4,1,3;2,4,5,3,1,2;4,3,1,2,5,3;1,2,5,3,4,1;4,3,4,1,2,3;2,1,2,3,4,1]
 [1,1,2,4,1,3;2,4,5,3,1,2;4,3,1,2,5,3;1,2,5,3,4,1;4,3,4,1,2,3;2,1,2,4,3,1]
 [1,1,2,4,1,3;2,5,4,3,1,2;4,3,1,2,3,4;1,2,3,4,5,1;3,4,5,1,2,3;2,1,2,3,4,1]
 [1,1,2,4,1,3;2,5,4,3,1,2;4,3,1,2,3,4;1,2,3,4,5,1;3,4,5,1,2,3;2,1,2,4,3,1]
 [1,1,2,4,1,3;2,5,4,3,1,2;4,3,1,2,3,4;1,2,3,4,5,1;3,5,4,1,2,3;2,1,2,3,4,1]
 [1,1,2,4,1,3;2,5,4,3,1,2;4,3,1,2,3,4;1,2,3,4,5,1;3,5,4,1,2,3;2,1,2,4,3,1]
 [1,1,2,4,1,3;2,5,4,3,1,2;4,3,1,2,3,4;1,2,3,5,4,1;3,4,5,1,2,3;2,1,2,3,4,1]
 [1,1,2,4,1,3;2,5,4,3,1,2;4,3,1,2,3,4;1,2,3,5,4,1;3,4,5,1,2,3;2,1,2,4,3,1]
 [1,1,2,4,1,3;2,5,4,3,1,2;4,3,1,2,3,4;1,2,3,5,4,1;3,5,4,1,2,3;2,1,2,3,4,1]
 [1,1,2,4,1,3;2,5,4,3,1,2;4,3,1,2,3,4;1,2,3,5,4,1;3,5,4,1,2,3;2,1,2,4,3,1]
 [1,1,2,4,1,3;2,5,4,3,1,2;4,3,1,2,4,3;1,2,4,3,5,1;4,3,5,1,2,3;2,1,2,3,4,1]
 [1,1,2,4,1,3;2,5,4,3,1,2;4,3,1,2,4,3;1,2,4,3,5,1;4,3,5,1,2,3;2,1,2,4,3,1]
 [1,1,2,4,1,3;2,5,4,3,1,2;4,3,1,2,4,3;1,2,5,3,5,1;4,3,4,1,2,3;2,1,2,3,4,1]
 [1,1,2,4,1,3;2,5,4,3,1,2;4,3,1,2,4,3;1,2,5,3,5,1;4,3,4,1,2,3;2,1,2,4,3,1]
 [1,1,2,4,1,3;2,5,4,3,1,2;4,3,1,2,5,3;1,2,4,3,4,1;4,3,5,1,2,3;2,1,2,3,4,1]
 [1,1,2,4,1,3;2,5,4,3,1,2;4,3,1,2,5,3;1,2,4,3,4,1;4,3,5,1,2,3;2,1,2,4,3,1]
 [1,1,2,4,1,3;2,5,4,3,1,2;4,3,1,2,5,3;1,2,5,3,4,1;4,3,4,1,2,3;2,1,2,3,4,1]
 [1,1,2,4,1,3;2,5,4,3,1,2;4,3,1,2,5,3;1,2,5,3,4,1;4,3,4,1,2,3;2,1,2,4,3,1]
 [1,2,3,1,3,2;3,4,5,2,4,1;2,1,1,3,5,2;3,4,2,4,1,3;1,5,3,5,2,4;3,2,1,1,3,1]
 [1,2,3,1,3,2;3,4,5,2,4,1;2,1,1,3,5,2;3,4,2,4,1,4;1,5,3,5,2,3;3,2,1,1,3,1]
 [1,2,3,1,3,2;3,4,5,2,4,1;2,1,1,3,5,2;3,4,2,5,1,3;1,5,3,4,2,4;3,2,1,1,3,1]
 [1,2,3,1,3,2;3,4,5,2,4,1;2,1,1,3,5,2;3,4,2,5,1,4;1,5,3,4,2,3;3,2,1,1,3,1]
 [1,2,3,1,3,2;3,4,5,2,4,1;2,1,1,3,5,2;3,5,2,4,1,3;1,4,3,5,2,4;3,2,1,1,3,1]
 [1,2,3,1,3,2;3,4,5,2,4,1;2,1,1,3,5,2;3,5,2,4,1,4;1,4,3,5,2,3;3,2,1,1,3,1]
 [1,2,3,1,3,2;3,4,5,2,4,1;2,1,1,3,5,2;3,5,2,5,1,3;1,4,3,4,2,4;3,2,1,1,3,1]
 [1,2,3,1,3,2;3,4,5,2,4,1;2,1,1,3,5,2;3,5,2,5,1,4;1,4,3,4,2,3;3,2,1,1,3,1]
 [1,2,3,1,3,2;3,4,5,2,5,1;2,1,1,3,4,2;3,4,2,4,1,3;1,5,3,5,2,4;3,2,1,1,3,1]
 [1,2,3,1,3,2;3,4,5,2,5,1;2,1,1,3,4,2;3,4,2,4,1,4;1,5,3,5,2,3;3,2,1,1,3,1]
 [1,2,3,1,3,2;3,4,5,2,5,1;2,1,1,3,4,2;3,4,2,5,1,3;1,5,3,4,2,4;3,2,1,1,3,1]
 [1,2,3,1,3,2;3,4,5,2,5,1;2,1,1,3,4,2;3,4,2,5,1,4;1,5,3,4,2,3;3,2,1,1,3,1]
 [1,2,3,1,3,2;3,4,5,2,5,1;2,1,1,3,4,2;3,5,2,4,1,3;1,4,3,5,2,4;3,2,1,1,3,1]
 [1,2,3,1,3,2;3,4,5,2,5,1;2,1,1,3,4,2;3,5,2,4,1,4;1,4,3,5,2,3;3,2,1,1,3,1]
 [1,2,3,1,3,2;3,4,5,2,5,1;2,1,1,3,4,2;3,5,2,5,1,3;1,4,3,4,2,4;3,2,1,1,3,1]
 [1,2,3,1,3,2;3,4,5,2,5,1;2,1,1,3,4,2;3,5,2,5,1,4;1,4,3,4,2,3;3,2,1,1,3,1]
 [1,2,3,1,3,2;3,5,4,2,4,1;2,1,1,3,5,2;3,4,2,4,1,3;1,5,3,5,2,4;3,2,1,1,3,1]
 [1,2,3,1,3,2;3,5,4,2,4,1;2,1,1,3,5,2;3,4,2,4,1,4;1,5,3,5,2,3;3,2,1,1,3,1]
 [1,2,3,1,3,2;3,5,4,2,4,1;2,1,1,3,5,2;3,4,2,5,1,3;1,5,3,4,2,4;3,2,1,1,3,1]
 [1,2,3,1,3,2;3,5,4,2,4,1;2,1,1,3,5,2;3,4,2,5,1,4;1,5,3,4,2,3;3,2,1,1,3,1]
 [1,2,3,1,3,2;3,5,4,2,4,1;2,1,1,3,5,2;3,5,2,4,1,3;1,4,3,5,2,4;3,2,1,1,3,1]
 [1,2,3,1,3,2;3,5,4,2,4,1;2,1,1,3,5,2;3,5,2,4,1,4;1,4,3,5,2,3;3,2,1,1,3,1]
 [1,2,3,1,3,2;3,5,4,2,4,1;2,1,1,3,5,2;3,5,2,5,1,3;1,4,3,4,2,4;3,2,1,1,3,1]
 [1,2,3,1,3,2;3,5,4,2,4,1;2,1,1,3,5,2;3,5,2,5,1,4;1,4,3,4,2,3;3,2,1,1,3,1]
 [1,2,3,1,3,2;3,5,4,2,5,1;2,1,1,3,4,2;3,4,2,4,1,3;1,5,3,5,2,4;3,2,1,1,3,1]
 [1,2,3,1,3,2;3,5,4,2,5,1;2,1,1,3,4,2;3,4,2,4,1,4;1,5,3,5,2,3;3,2,1,1,3,1]
 [1,2,3,1,3,2;3,5,4,2,5,1;2,1,1,3,4,2;3,4,2,5,1,3;1,5,3,4,2,4;3,2,1,1,3,1]
 [1,2,3,1,3,2;3,5,4,2,5,1;2,1,1,3,4,2;3,4,2,5,1,4;1,5,3,4,2,3;3,2,1,1,3,1]
 [1,2,3,1,3,2;3,5,4,2,5,1;2,1,1,3,4,2;3,5,2,4,1,3;1,4,3,5,2,4;3,2,1,1,3,1]
 [1,2,3,1,3,2;3,5,4,2,5,1;2,1,1,3,4,2;3,5,2,4,1,4;1,4,3,5,2,3;3,2,1,1,3,1]
 [1,2,3,1,3,2;3,5,4,2,5,1;2,1,1,3,4,2;3,5,2,5,1,3;1,4,3,4,2,4;3,2,1,1,3,1]
 [1,2,3,1,3,2;3,5,4,2,5,1;2,1,1,3,4,2;3,5,2,5,1,4;1,4,3,4,2,3;3,2,1,1,3,1]